L(s) = 1 | − 0.219·2-s + 3-s − 1.95·4-s + (1.49 + 1.65i)5-s − 0.219·6-s + (2.26 + 1.36i)7-s + 0.865·8-s + 9-s + (−0.328 − 0.363i)10-s + (−3.31 + 0.00596i)11-s − 1.95·12-s + 4.76i·13-s + (−0.495 − 0.300i)14-s + (1.49 + 1.65i)15-s + 3.71·16-s + 1.75i·17-s + ⋯ |
L(s) = 1 | − 0.154·2-s + 0.577·3-s − 0.976·4-s + (0.670 + 0.742i)5-s − 0.0894·6-s + (0.855 + 0.517i)7-s + 0.306·8-s + 0.333·9-s + (−0.103 − 0.114i)10-s + (−0.999 + 0.00179i)11-s − 0.563·12-s + 1.32i·13-s + (−0.132 − 0.0802i)14-s + (0.386 + 0.428i)15-s + 0.928·16-s + 0.424i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483025435\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483025435\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (-1.49 - 1.65i)T \) |
| 7 | \( 1 + (-2.26 - 1.36i)T \) |
| 11 | \( 1 + (3.31 - 0.00596i)T \) |
good | 2 | \( 1 + 0.219T + 2T^{2} \) |
| 13 | \( 1 - 4.76iT - 13T^{2} \) |
| 17 | \( 1 - 1.75iT - 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 + 5.32iT - 23T^{2} \) |
| 29 | \( 1 - 7.93iT - 29T^{2} \) |
| 31 | \( 1 + 0.416iT - 31T^{2} \) |
| 37 | \( 1 + 8.40iT - 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 - 4.16iT - 53T^{2} \) |
| 59 | \( 1 + 1.47iT - 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 - 14.7iT - 73T^{2} \) |
| 79 | \( 1 - 4.99iT - 79T^{2} \) |
| 83 | \( 1 + 4.77iT - 83T^{2} \) |
| 89 | \( 1 + 8.37iT - 89T^{2} \) |
| 97 | \( 1 - 8.91T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03547099072318899613677254749, −8.890097125023152610153689960966, −8.731030297173100811215495549927, −7.71494842252103887177232270598, −6.79620071407619423880872187186, −5.70247062677106403987719613663, −4.82951694385794402932722620345, −3.95797845180738624860154927741, −2.61063828882126409617179351226, −1.75051782782015836420621394304,
0.65242293347394741889724321246, 1.96781616272608413737830398609, 3.33413077090768791257249843140, 4.56958541269574305346211310499, 5.05864103209392613148829990760, 5.94806015559597419447798583332, 7.53328105333342104178933914121, 8.160065186348469520960308838747, 8.535250877337870831183868373255, 9.676057178078815968212726182115